The California-Kepler Survey. IV. Metal-rich Stars Host a Greater Diversity of Planets

CKS is a large-scale spectroscopic survey of 1305 Kepler Objects of Interest (KOIs). The sample selection, spectroscopic observations, and spectroscopic analysis are described in detail in Petigura et al. (2017b, hereafter Paper I). In brief, the sample was initially constructed by selecting all KOIs brighter than Kp = 14.2 mag.⁹ A KOI is a Kepler target star which showed periodic photometric dimmings indicative of planet transits. However, not all KOIs have received the necessary follow-up attention needed to confirm the planets.

Over the course of the project, we included additional targets to cover different planet populations, including multi-candidate hosts, Ultra Short Period candidates, and Habitable Zone candidates. We obtained spectra at the 10 m Keck Telescope using the High Resolution Echelle Spectrometer at a resolution of R = 60,000 (Vogt et al. 1994). A key feature of the CKS data set is that stars were observed to a consistent signal-to-noise ratio (S/N) of 45 pixel⁻¹ at the peak of the blaze function near 5500 Å.

In Paper I, we extracted the following properties for each star: effective temperature ${T}_{\mathrm{eff}}$, surface gravity $\mathrm{log}g$, metallicity $[\mathrm{Fe}/{\rm{H}}]$, and projected rotational velocity $v\sin i$ using the SpecMatch (Petigura 2015) and SME@XSEDE codes (Paper I).

In Johnson et al. (2017, hereafter Paper II), we converted the spectroscopic properties of ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, and $[\mathrm{Fe}/{\rm{H}}]$ into stellar mass ${M}_{\star }$, radius ${R}_{\star }$, and age. This conversion is facilitated by the publicly available isochrones Python package (Morton 2015). Stellar mass, radius, and age are measured to 4%, 11%, and 30%, respectively. We then recompute planetary radii and equilibrium temperatures using our updated CKS parameters and the results from transit fitting performed by Mullally et al. (2015). Paper II provides updated stellar and planetary properties for 1305 KOIs and 2025 planet candidates, which are the starting point for our planet sample ${ \mathcal P }$.

We applied a series of filters, described below in bullets, to the CKS sample to arrive at our final planet sample ${ \mathcal P }$. The filters restrict the range of stellar properties included in our analysis. Since we aim to explore the connection between metallicity and planet size, we also require the planets have well-determined radii.

Where possible, we filter based on the DR25 stellar properties table of Mathur et al. (2017). These cuts may be applied homogeneously to the field star sample ${ \mathcal S }$. Some filters involve follow-up observations not available for every star in ${ \mathcal S }$. In Section 2.3, we quantify the number of field stars that would have been excluded had comparable follow-up been performed. The applied filters are as follows:

• 1.  
  Stellar brightness. We restrict our sample to the magnitude-limited sub-sample of the CKS sample (i.e., ${Kp}\lt 14.2\,\mathrm{mag}$).
• 2.  
  Stellar effective temperature. The spectroscopic tools used in Paper I produce reliable results for ${T}_{\mathrm{eff}}$ = 4700–6500 K. We restrict our analysis to stars having photometric ${T}_{\mathrm{eff}}$ = 4700–6500 K (DR25 stellar properties table).
• 3.  
  Stellar surface gravity. We restrict our analysis to stars having photometric $\mathrm{log}g$ = 3.9–5.0 dex (DR25 stellar properties table).
• 4.  
  Stellar dilution. Dilution from nearby stars can also alter the apparent planetary radii. Furlan et al. (2017) compiled high resolution imaging observations performed by several groups.¹⁰ When a nearby star is detected, Furlan et al. (2017) computed a radius correction factor (RCF), which accounts for dilution assuming the planet transits the brightest star. We elect to not apply this correction factor, but conservatively exclude KOIs where the RCF is larger than 5%.
• 5.  
  Planet orbital period. We remove planet candidates with orbital periods longer than 350 days. This excludes planets that only transit once or twice during Kepler observations, which have a higher false positive rate (Mullally et al. 2015).
• 6.  
  Planet false positive designation. We exclude candidates that are identified as false positives according to Paper I.
• 7.  
  Planets with grazing transits. Finally, we exclude stars having grazing transits (b > 0.9), which have suspect radii due to covariances with the planet size and stellar limb-darkening during the light curve fitting.

Our successive filters, along with a running tally of the number planets which pass them, are listed in Table 1. The filters on dilution and grazing transits each eliminate a small percentage (8% and 5%) of planets with imprecise radii, even after CKS spectroscopy. Similar filters clarified the bimodal planet radius distribution presented by Fulton et al. (2017), Paper III, in the CKS series.

Table 1.  Filters Applied to Planet Sample

Filter	${n}_{\mathrm{pl},\mathrm{pass}}$	${n}_{\mathrm{pl},\mathrm{pass},\mathrm{run}}$	${f}_{\mathrm{pl},\mathrm{pass},\mathrm{run}}$	${n}_{\star ,\mathrm{pass},\mathrm{run}}$
Full sample	2025	2025	1.000	1279
Kp < 14.2 mag	1359	1359	0.671	954
${T}_{\mathrm{eff}}$ = 4700–6500 K	1977	1328	0.977	929
$\mathrm{log}g=3.9\mbox{--}5.0\,\mathrm{dex}$	1899	1212	0.913	829
P < 350 days	1965	1191	0.983	817
Not a false positive	1861	1105	0.928	758
Radius correction factor <5%	1891	1017	0.920	695
Not a grazing transit (b < 0.9)	1870	970	0.954	662

Note. Summary of the filters applied to the CKS catalog to create planet sample ${ \mathcal P }$. The column labeled ${n}_{\mathrm{pl},\mathrm{pass}}$ is the total number of KOIs that pass a specific filter and ${n}_{\mathrm{pl},\mathrm{pass},\mathrm{run}}$ is a running tally of KOIs that pass all filters. For filter i, ${f}_{\mathrm{pl},\mathrm{pass},\mathrm{run}}(i)\,={n}_{\mathrm{pl},\mathrm{pass},\mathrm{run}}(i)/{n}_{\mathrm{pl},\mathrm{pass},\mathrm{run}}(i-1)$. For example, ${f}_{\mathrm{pl},\mathrm{pass},\mathrm{run}}(3)\,={n}_{\mathrm{pl},\mathrm{pass},\mathrm{run}}(3)/{n}_{\mathrm{pl},\mathrm{pass},\mathrm{run}}(2)\,=\,1328/1359\,=\,0.977$. The column labeled ${n}_{\star ,\mathrm{pass},\mathrm{run}}$ gives the numbers of unique stars hosting the ${n}_{\mathrm{pl},\mathrm{pass},\mathrm{run}}$ KOIs.

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In total, ${ \mathcal P }$ contains  970  planets orbiting  662  stars that pass all filters. Figure 1 shows planets on the $[\mathrm{Fe}/{\rm{H}}]$–${R}_{P}$ plane as successive cuts are applied. Even though planets across this plane are filtered out, planets larger than Neptune have a higher rate of removal. Properties of the stars hosting these planets are shown in Figure 2.

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In order to compare the properties of stars with and without transiting planets, we need to consider the parent population of Kepler field stars ${ \mathcal S }$, from which ${ \mathcal P }$ is drawn. We begin with the Mathur et al. (2017) catalog that lists all 199,991  stars observed at some point during the Kepler mission. Where possible, we apply the same set of filters to construct ${ \mathcal S }$ that were used to construct ${ \mathcal P }$. Table 2 summarizes the number of stars that pass cuts on ${Kp}$, photometric ${T}_{\mathrm{eff}}$, and photometric $\mathrm{log}g$. A total of  36,959  stars pass all cuts. The distribution of field star properties is shown in Figure 2.

Table 2.  Filters Applied to Stellar Sample

Cut	${n}_{\star ,{\rm{pass}}}$	${n}_{\star ,\mathrm{pass},\mathrm{run}}$	${f}_{\star ,\mathrm{pass},\mathrm{run}}$
Full sample	199991	199991	1.000
Kp < 14.2 mag	81758	81758	0.409
${T}_{\mathrm{eff}}$ = 4700–6500 K	168885	62751	0.768
$\mathrm{log}g=3.9\mbox{--}5.0\,\mathrm{dex}$	162854	36959	0.589

Note. Summary of the filters applied to ${ \mathcal S }$. See Table 1 for column descriptions.

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The filters used to construct ${ \mathcal P }$ were not simply cuts on the stellar properties, but also on quality of the CKS stellar radii and the properties of the planet candidates. We cannot apply these same filters to the field stars on a star-by-star basis because these stars do not have CKS stellar radii or detected planets. However, we must assess whether the remaining filters from Section 2.2 would have excluded a significant fraction of field stars, assuming all stars received similar follow-up attention. Here, we quantify the number of field stars that would have been excluded from ${ \mathcal S }$ had comparable follow-up been performed.

• 1.  
  Reliable spectroscopic parameters. Our initial sample of planets orbit stars for which the spectroscopic analysis described in Section 2 produced reliable results. For a star to be included, $v\sin i$ must be less than 20 km s⁻¹. This selection effect excludes some stars that are typically near 6500 K, where $v\sin i$ values begin to exceed 20 km s⁻¹. After filtering on ${Kp}$, photometric ${T}_{\mathrm{eff}}$, and photometric $\mathrm{log}g$, 3.1% of stars are excluded because they do not have reliable spectroscopic parameters.
• 2.  
  Stellar dilution. 7.8% of planet candidates were excluded because the Kepler apertures contained enough flux from neighboring stars, such that the planet radii required a correction factor larger than 5%. The fraction of stars excluded depends on the crowding of Kepler field stars. We make the assumption that this distribution is independent of KOI status. Had all field stars received a similar level of high-contrast imaging follow-up, 7.8% would have companions sufficiently bright and close to warrant a RCF of $\gt 5 \%$.
• 3.  
  Planet orbital period. For a KOI to be included in our sample, we required that the orbital period be less than 350 days. Such a cut is strictly a cut on the planet properties and does not preclude any stars from being included in ${ \mathcal S }$.
• 4.  
  Planet false positive disposition. A star must first be identified as a KOI in order to then be designated as a false positive. Therefore, the false positive filter does not exclude significant numbers of field stars from ${ \mathcal S }$.
• 5.  
  Planets with grazing transits. We require that the planets have non-grazing impact parameters. As with the radius filter, it does not affect inclusion in ${ \mathcal S }$. However, this cut on impact parameter must be factored into the geometric transit probability, which affects the occurrence calculation described in Section 4.

When we account for stars that would have been excluded from our sample due to unreliable spectroscopic parameters or the presence of nearby stars, we find that the planet sample ${ \mathcal P }$ was drawn from a parent stellar sample ${ \mathcal S }$ containing 36,959 × 0.922 × 0.969 = 33,020 stars.

Here, we characterize the metallicity distribution of the parent sample ${ \mathcal S }$. While the KIC (Brown et al. 2011) and its updates (e.g., Mathur et al. 2017) tabulate metallicities for every Kepler target star, they are inadequate for characterizing the metallicity distribution of the Kepler field given their low precision. Instead, we use the LAMOST/LASP stellar parameters (Luo et al. 2015) from Data Release 3 (DR3).¹¹ Following De Cat et al. (2015), we crossmatch the LAMOST-DR3 with the KIC by identifying LAMOST spectra taken within 1.2 arcsec of a KIC target star. This crossmatching results in  29,997  stars in common having ${Kp}$ < 14.2 mag. We note a systematic offset between the LAMOST and CKS metallicity scales of ≈0.04 dex when comparing  476  stars in common. We apply a correction, described in Appendix A, but estimate that residual systematic offsets of ≈0.01 dex remain.

We apply the same set of filters to the LAMOST stars that we applied to ${ \mathcal P }$. These include cuts in ${Kp}$, ${T}_{\mathrm{eff}}$, and $\mathrm{log}g$. After applying these cuts, we are left with  14,382  stars with LAMOST parameters. The filters are summarized in Table 3 and the distribution of LAMOST stellar properties is shown in Figure 2.

Table 3.  Summary of Cuts to LAMOST Sample

Cut	${n}_{\star ,\mathrm{pass}}$	${n}_{\star ,\mathrm{pass},\mathrm{run}}$	${f}_{\star ,\mathrm{pass},\mathrm{run}}$
Full sample	29997	29997	1.000
Kp < 14.2 mag	29997	29997	1.000
${T}_{\mathrm{eff}}$ = 4700–6500 K	23551	23551	0.785
$\mathrm{log}g=3.9\mbox{--}5.0\,\mathrm{dex}$	18625	14382	0.611

Note. Summary of the cuts applied to the LAMOST DR-2 sample. See Table 1 for column descriptions.

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Table 4 summarizes the metallicity distribution of ${ \mathcal P }$ and ${ \mathcal S }$ measured through different methods. The mean metallicity of ${ \mathcal S }$, as measured by LAMOST, is −0.01 dex, similar to the mean metallicity of the filtered planet sample ${ \mathcal P }$, +0.03 dex, as measured by CKS. We note that the mean metallicity of ${ \mathcal S }$, according to the DR25 stellar properties table (Mathur et al. 2017), is −0.19 dex. This low value is due to the low metallicity prior used in the photometric modeling.

Table 4.  Comparison of Planet Host and Field Star Metallicities

	${ \mathcal P }$ (spec)	${ \mathcal S }$ (phot)	${ \mathcal S }$ (spec)
	(dex)	(dex)	(dex)
Mean	0.031	−0.187	−0.005
rms	0.185	0.259	0.207
SEM	0.006	0.001	0.002
25%	−0.069	−0.320	−0.116
50%	0.052	−0.160	0.020
75%	0.148	−0.020	0.131

Note. Summary of metallicity distributions for different stellar samples. ${ \mathcal P }$ (spec) refers to the CKS metallicities of our filtered list of planet hosts (Section 2.2). ${ \mathcal S }$ (phot) refers to the photometric metallicities of our parent stellar sample (Section 2.3). ${ \mathcal S }$ (spec) refers to the metallicities of ${ \mathcal S }$ measured using LAMOST data (Section 2.4). ${ \mathcal S }$ (phot) and ${ \mathcal S }$ (spec) have significantly different mean metallicities, due the metallicity prior imposed by Mathur et al. (2017).

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In Figure 3, we show the planet sizes and host star metallicities for the 2025 planet candidates in the CKS sample. Planets smaller than Neptune are found around stars of wide-ranging metallicities, while there is a deficit of planets larger than Neptune around stars with sub-solar metallicity.

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To investigate variation in average host star metallicities with planet sizes, we divided the metallicity measurements according to planet size. Bins span a factor of $\sqrt{2}$ in ${R}_{P}$ for planets smaller than 4 ${R}_{\oplus }$. Larger planets are placed into bins spanning a factor of 2, owing to lower numbers. Figure 3 shows the mean metallicities for these planet radius bins along with the 25% and 75% quantiles. We observe a gradual upward trend in mean host star metallicity from smaller to larger planets. Buchhave et al. (2012, 2014) observed a similar trend in smaller samples of planet hosts.

Figure 3 also shows host star metallicity as a function of orbital period. Unlike the ${R}_{P}$–$[\mathrm{Fe}/{\rm{H}}]$, there are no large regions of the P–$[\mathrm{Fe}/{\rm{H}}]$ plane clearly devoid of planets. After computing the mean metallicity in bins of P spanning 0.25 dex, we observe a slight increase in mean metallicity of about 0.05 dex with decreasing orbital period over P = 1–10 days. Mulders et al. (2016) observed a similar trend in a smaller sample of planet hosts.

Here, we examine the effect of host star metallicity on the 2D distribution of planet size and orbital period. We divided ${ \mathcal P }$ into four bins of host star metallicity with boundaries at  −0.116,  +0.020, and  +0.131 dex. The boundaries of the bins equally divide the stars in ${ \mathcal S }$ (see Table 4). Figure 4 shows the distribution of planets in the P–${R}_{P}$ plane for the different metallicity bins. If the occurrence of planets were independent of host star metallicity, then the distribution of planets in each plot would be indistinguishable. The panels of Figure 4 show clear differences, indicating that metallicity is associated with certain types of planets, and that the strength of that enhancement depends on both P and ${R}_{P}$.

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To facilitate our investigation into the effect of metallicity across the P–${R}_{P}$ plane, we define several regions. As a matter of convenience, we define a nomenclature for referring to these different regions. While the boundaries are matters of taste, we choose physically motivated boundaries, when possible. We consider four domains of planet size, defined below:

• 1.  
  Jupiters. ${R}_{P}$ = 8–24 ${R}_{\oplus }$. The lower limit is motivated by the fact that planets larger than 8 ${R}_{\oplus }$ tend to have masses ranging from 100 to 10,000 ${M}_{\oplus }$, while planets smaller than 8 ${R}_{\oplus }$ tend to have lower masses ranging from 6 to 60 ${M}_{\oplus }$ (see, e.g., Petigura et al. 2017b, Figure 9). At ${M}_{P}$ ≈ 100 ${M}_{\oplus }$ electron degeneracy pressure begins contribute significantly to a planet's pressure support (Zapolsky & Salpeter 1969). Thus 8 ${R}_{\oplus }$ approximates the dividing line between planets with different pressure support and interior structures. The upper radius limit of 24 ${R}_{\oplus }$ is based on the known sizes of hot Jupiters. The most highly irradiated Jovians discovered to date can exceed 2 ${R}_{\mathrm{jup}}$ (e.g., WASP-79b; Smalley et al. 2012).^{12 ,13}
• 2.  
  Sub-Saturns. ${R}_{P}$ = 4–8 ${R}_{\oplus }$. Sub-Saturns are typically typically less massive the larger Jupiters. They are roughly 10× more rare than the smaller sub-Neptunes, suggesting a different formation pathway. A radius of 4 ${R}_{\oplus }$ approximates a breakpoint in the planet size distribution where planet occurrence rises rapidly with decreasing size (see Fulton et al. 2017).
• 3.  
  Sub-Neptunes. ${R}_{P}$ = 1.7– 4.0 ${R}_{\oplus }$. The lower radius limit corresponds to a likely transition radius between rocky planets and planets that have envelopes that are an appreciable fraction of the total planet size. Among the observations supporting such a transition are measurements of planet bulk density from transits and radial velocities (Marcy et al. 2014; Weiss & Marcy 2014; Rogers 2015) and the observation of a gap in the radius distribution of planets by Fulton et al. (2017).
• 4.  
  Super-Earths. ${R}_{P}$ = 1.0–1.7 ${R}_{\oplus }$. Planets that are smaller than sub-Neptunes. Few planets in this size range have well-measured masses, but those that do are often consistent with rocky compositions.

We also consider three domains of orbital period:

• 1.  
  Hot Planets. P = 1–10 days. The upper limit of 10 days corresponds to a breakpoint in distribution of planet occurrence with orbital period (Howard et al. 2012; Fressin et al. 2013). Below $P\approx 10\,\mathrm{days}$, planet occurrence per $\mathrm{log}P$ is observed to decline, while planet occurrence is roughly constant for longer periods.
• 2.  
  Warm Planets. P = 10–100 days. An intermediate range of orbital periods. While warm planets are intrinsically more common than hot planets, they represent about half of the total sample due to falling completeness and transit probability with increasing orbital period.
• 3.  
  Cool Planets. $P=100\mbox{--}350\,\mathrm{days}$. The longest period planets included in our survey. There are very few planets in our sample with such long periods due to falling completeness and decreasing transit probability.

Here, we note some qualitative features in Figure 4. Stars of all metallicity bins host warm super-Earths and warm sub-Neptunes. Cool Jupiters are intrinsically rare at all metallicities, but they are present in all four metallicity bins. We observe a steady increase of hot Jupiters with increasing metallicity. Sub-Saturns of all orbital periods are almost completely absent in the lower two metallicity bins, which represent half of the parent sample ${ \mathcal S }$; they are almost exclusively found around high metallicity stars. While there are some examples of hot super-Earths in each metallicity bin, their numbers increase with increasing metallicity. Finally, there is almost a complete absence of hot sub-Neptunes in the lowest metallicity bins. Hot sub-Neptunes are more common with increasing metallicity.

In order to quantitatively assess the extent to which metallicity enhances the production of different types of planets, we compare the distribution of planet host metallicities to that of the field star population. In Table 5, we list the mean metallicities of the various planet subclasses as well as the standard error of the mean (SEM), which we compare to the mean field star metallicity, as measured from LAMOST spectra.

Table 5.  Comparison of Planet Host and Field Star Metallicities

Name	P	${R}_{P}$	${n}_{\star }$	$\langle [\mathrm{Fe}/{\rm{H}}]\rangle$	t-stat	p-value	Siga
	day	${R}_{\oplus }$		dex
Hot Jupiters	1–10	8.0–24.0	14	+0.19 ± 0.04	4.5	<8 × 10−4	Y
Warm Jupiters	10–100	8.0–24.0	4	+0.06 ± 0.02	3.4	<6 × 10−2	N
Cool Jupiters	100–350	8.0–24.0	13	+0.06 ± 0.05	1.4	<3 × 10−1	N
Hot Sub-Saturns	1–10	4.0–8.0	7	+0.26 ± 0.04	6.5	<7 × 10−4	Y
Warm Sub-Saturns	10–100	4.0–8.0	19	+0.14 ± 0.02	5.6	<6 × 10−5	Y
Cool Sub-Saturns	100–350	4.0–8.0	7	+0.11 ± 0.06	1.9	<1 × 10−1	N
Hot Sub-Neptunes	1–10	1.7–4.0	108	+0.11 ± 0.02	7.0	<4 × 10−9	Y
Warm Sub-Neptunes	10–100	1.7–4.0	282	+0.04 ± 0.01	4.0	<2 × 10−3	Y
Cool Sub-Neptunes	100–350	1.7–4.0	29	−0.05 ± 0.03	−1.3	<3 × 10−1	N
Hot Super-Earths	1–10	1.0–1.7	181	+0.05 ± 0.01	4.7	<2 × 10−4	Y
Warm Super-Earths	10–100	1.0–1.7	132	−0.04 ± 0.02	−2.2	<1 × 10−1	N
Cool Super-Earths	100–350	1.0–1.7	4	−0.36 ± 0.03	−11.4	<2 × 10−3	Y
All Jupiters	1–350	8.0–24.0	31	+0.12 ± 0.03	4.2	<6 × 10−4	Y
All Sub-Saturns	1–350	4.0–8.0	33	+0.16 ± 0.02	7.2	<1 × 10−7	Y
All Sub-Neptunes	1–350	1.7–4.0	419	+0.05 ± 0.01	6.2	<7 × 10−7	Y
All Super-Earths	1–350	1.0–1.7	317	+0.01 ± 0.01	1.3	<8 × 10−1	N

Note. We inspected the metallicity distribution of various groups of planets defined by their sizes and orbital periods. For each group, we computed the mean metallicity $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ and the standard error of the mean. We compare these metallicities to the metallicities of Kepler field stars, as measured by LAMOST, using the student t-test. This test returns a t-statistic, which quantifies the statistical difference in mean metallicities, and a p-value, the probability that the two samples were drawn from distributions with the same mean.

^aIs the difference between planet host and field star metallicities significant (i.e., is $p \mbox{-} \mathrm{value}$ < 10⁻²)?

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We assess the significance of the difference between field star and planet host star metallicities using the student t-test, which evaluates the difference between the means of the two samples in units of SEM, the t-statistic. The t-test also returns a $p \mbox{-} \mathrm{value}$, which is the probability that field stars and planet host stars are drawn from distributions with the same mean value.

We may observe statistically significant differences in mean metallicity for two reasons: (1) intrinsic differences between planet host and field star metallicities or (2) residual offsets in the CKS and LAMOST metallicity scales. While we calibrated LAMOST metallicities to the CKS scale, we estimate that offsets of 0.01 dex may remain (see Section 2.4 and Appendix A). To account for the possibility of such residual offsets, we perform three t-tests for each sample where we shift the LAMOST metallicities by −0.01, 0.00, and +0.01 dex. Each of these different tests returns a different $p \mbox{-} \mathrm{value}$. We use the largest (most conservative) $p \mbox{-} \mathrm{value}$ to assess differences in mean metallicities. If the largest p-value is less than 0.01, we deem the metallicity difference to be significant.

Stars hosting Jupiter-size planets have enhanced metallicities, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ =  +0.12  ±  0.04 dex. The hot Jupiters hosts, with mean metallicity of $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ =  +0.19  ±  0.04 dex, are significantly enhanced relative to field stars ($p \mbox{-} \mathrm{value}$ <  $8\times {10}^{-4}$). While the mean metallicities for the warm and cool Jupiter hosts are also enhanced, the small numbers of such planets prevent a high significance detection of a metallicity enhancement.

Of all the planet size classes studied, the sub-Saturn hosts have the highest mean metallicity, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ = +0.16 ± 0.02 dex. Among the sub-Saturns, the hot sub-Saturns have the highest mean host star metallicity, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ =  +0.26  ± 0.04 dex. We find that the hot and warm sub-Saturns hosts were significantly enhanced compared to field stars, while small numbers of detected cool sub-Saturns prevented a detailed comparison.

As a whole, the sub-Neptunes have a mean metallicity of $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ =  +0.05  ±  0.01 dex, close to the field star value. However, when split according to orbital period, we find that the hot sub-Neptunes have an enhanced mean metallicity, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ =  +0.11  ±  0.02 dex. The large mean metallicity, combined with the large number of such planets (N =  108), results in a very significant detection of a metallicity enhancement ($p \mbox{-} \mathrm{value}$ < $4\times {10}^{-9}$). Due to their high detectability and high intrinsic occurrence, the warm sub-Neptunes have the largest total number of any P–${R}_{P}$ subclass, N =  282. The large number of planets means that even small offsets in mean metallicities can be significant. While the mean metallicity of warm sub-Neptunes, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ =  +0.04  ±  0.01 dex), was not as high as that of the hot sub-Neptunes, it was significantly elevated relative to field stars ($p \mbox{-} \mathrm{value}$ < $2\times {10}^{-3}$).

Finally, the super-Earth-size planets have the lowest mean metallicity of  +0.01  ±  0.01 dex, which is consistent with field star metallicity. However, as with the sub-Neptunes, the hot super-Earth hosts exhibit enhanced metallicity of  +0.05 ± 0.01 dex. The difference in mean metallicity (in dex) is not as large for the sub-Neptunes, so while the offset is still significant ($p \mbox{-} \mathrm{value}$ <  $2\times {10}^{-4}$ ), it is not as significant as for the hot sub-Neptunes. Despite the large number of warm super-Earths, we cannot detect a significant offset in mean metallicity.

One may wonder whether the mean metallicities of stars hosting warm super-Earths/sub-Neptunes could possibly be different from the mean metallicity of the Kepler field (−0.01 dex), given the high intrinsic occurrence of these planets. While numerous previous works have shown that warm super-Earths and sub-Neptunes are intrinsically common (see, e.g., Petigura 2013), neither class of planets is found around 100% of stars. For example, in Section 5 we show that there are $\approx 17$ warm super-Earths per 100 stars. Therefore, the mean metallicity of warm super-Earth hosts could be significantly different than that of field stars. We consider the following limiting case: Suppose that the process that produces warm super-Earths has a step function dependence on metallicity. Among Kepler targets, 17% of stars have $[\mathrm{Fe}/{\rm{H}}]$ > 0.16 dex. A universe where every star with $[\mathrm{Fe}/{\rm{H}}]$ > 0.16 dex produced one warm super-Earth, and every star with $[\mathrm{Fe}/{\rm{H}}]$ < 0.16 dex produced zero, would be consistent with the measured occurrence. By consulting the distribution of Kepler field star metallicities (see Section 2.4), we find that the average metallicity of all stars with $[\mathrm{Fe}/{\rm{H}}]$ > 0.16 dex is 0.25 dex. In this limiting case, the average metallicity of warm super-Earth hosts would be $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ = 0.25 dex, much higher than the observed value of  −0.04  ±  0.02 dex. That warm super-Earth and sub-Neptune hosts have mean metallicities nearly the same as that of field stars demonstrates qualitatively that their occurrence is not a steep function of metallicity. We show this quantitatively in Section 6.

In summary, close-in ($P\lt 10\,\mathrm{days}$) planets of all sizes have enhanced metallicity host stars. Warm super-Earth-size planets at intermediate distances (P = 10–100 days) orbit stars with a metallicity distribution similar to that of field stars. Larger planets in this period range have enhanced metallicities, and the sub-Neptunes and sub-Saturns have a significant metallicity enhancement. Finally, the number of detected cool planets (P = 100–350 days) of all sizes is too small to permit the detection of different host star metallicities.

Adopting the notation of Youdin (2011), the probability that a star with properties ${\boldsymbol{z}}$ has a planet with properties ${\boldsymbol{x}}$ that lie in a volume of $d{\boldsymbol{x}}$ of phase space is

$$\begin{eqnarray}&&{df}=\displaystyle \frac{\partial f({\boldsymbol{x}},{\boldsymbol{z}})}{\partial {\boldsymbol{x}}}d{\boldsymbol{x}}.\end{eqnarray} \tag{ 1 }$$

In this paper, ${\boldsymbol{z}}$ is metallicity M = $[\mathrm{Fe}/{\rm{H}}]$ ^,14 and ${\boldsymbol{x}}$ is some combination of $\mathrm{log}P$ and $\mathrm{log}{R}_{P}$.¹⁵ As a matter of convenience we express differential distribution as

$$\begin{eqnarray}&&\displaystyle \frac{\partial f({\boldsymbol{x}},{\boldsymbol{z}})}{\partial {\boldsymbol{x}}}={Cg}({\boldsymbol{x}},{\boldsymbol{z}}),\end{eqnarray} \tag{ 2 }$$

where C is a normalization constant and g is a shape function that may depend on stellar and/or planetary properties.

The number of planets per star (NPPS) having stellar properties ${\boldsymbol{z}}$ over a specified volume of planet properties X is

$$\begin{eqnarray}&&f({\boldsymbol{z}})={\int }_{X}{Cg}({\boldsymbol{x}},{\boldsymbol{z}})d{\boldsymbol{x}}.\end{eqnarray} \tag{ 3 }$$

The NPPS having stellar properties within a range of stellar properties Z is

$$\begin{eqnarray}&&f=\displaystyle \frac{1}{{n}_{\star }}\displaystyle \sum _{j}{\int }_{X}{Cg}({\boldsymbol{x}},{{\boldsymbol{z}}}_{j})d{\boldsymbol{x}},\end{eqnarray} \tag{ 4 }$$

where j labels stars where ${z}_{j}\in Z$.

Note that our definition of planet occurrence is NPPS. As a matter of convenience, we often express this in units of planets per 100 stars. Some papers (e.g., Fischer & Valenti 2005) define planet occurrence as the fraction of stars with planets (FSWP). For a transit survey like Kepler, computing FSWP is challenging because one must compute the number of stars without planets. However, a transit survey cannot distinguish between stars without planets and stars without transiting planets. Converting between NPPS and FSWP requires detailed modeling of planet multiplicity and mutual inclinations and is beyond the scope of this work. For more discussion on these points, see Youdin (2011) and references therein.

In this paper, we compute and display planet occurrence over rectilinear "cells" constructed from various combinations of $\mathrm{log}P$, $\mathrm{log}{R}_{P}$, and M. We express the size of these cells as ${\rm{\Delta }}\mathrm{log}P$, ${\rm{\Delta }}\mathrm{log}{R}_{P}$, ${\rm{\Delta }}M$, measured in dex. The occurrence within a cell is ${f}_{\mathrm{cell}}$, which depends on the number of independent trials ${n}_{\mathrm{trial}}$ that could have yielded a detected planet. In the limit where every star in ${ \mathcal S }$ has one transiting planet that is also detectable by the Kepler pipeline, ${n}_{\mathrm{trial}}$ = ${n}_{\star }$. In practice, ${n}_{\mathrm{trial}}$ must account for (1) non-transiting orbital inclinations and (2) lack of detectability due to insufficient S/N. For a given planet size and orbital period,

$$\begin{eqnarray}&&{n}_{\mathrm{trial}}=\displaystyle \sum _{i=1}^{{n}_{\star }}\,{p}_{\mathrm{tr},i}(P){p}_{\det ,i}(P,{R}_{P}),\end{eqnarray} \tag{ 5 }$$

where i labels each star in ${ \mathcal S }$, ${p}_{\mathrm{tr},i}$ is the transit probability, and ${p}_{\det ,i}$ is the probability that a transiting planet would be detectable by the Kepler pipeline. We represent this more compactly as

$$\begin{eqnarray}&&{n}_{\mathrm{trial}}={n}_{\star }\langle {p}_{\mathrm{tr},i}\,{p}_{\det ,i}\rangle ,\end{eqnarray} \tag{ 6 }$$

where $\langle \cdot \rangle$ denotes the arithmetic mean.

Following Bowler et al. (2015), if we assume that planet occurrence is log-uniform within a cell, ${n}_{\mathrm{trial}}$ for a cell of size ${\rm{\Delta }}{\boldsymbol{x}}={\rm{\Delta }}\mathrm{log}P\times {\rm{\Delta }}\mathrm{log}{R}_{P}$ is

$$\begin{eqnarray}&&{n}_{\mathrm{trial}}=\displaystyle \frac{{n}_{\star }}{{\rm{\Delta }}{\boldsymbol{x}}}\int \langle {p}_{\mathrm{tr}}{p}_{\det }\rangle d{\boldsymbol{x}}.\end{eqnarray} \tag{ 7 }$$

For a given cell with ${n}_{\mathrm{trial}}$ trials, ${n}_{\mathrm{pl}}$ detected planets, and ${n}_{\mathrm{nd}}$ = ${n}_{\mathrm{trial}}$–${n}_{\mathrm{pl}}$ non-detections, the likelihood of ${f}_{\mathrm{cell}}$ is given by the binomial distribution

$$\begin{eqnarray}&&P({f}_{\mathrm{cell}}| {n}_{\mathrm{pl}},{n}_{\mathrm{trial}})=P({n}_{\mathrm{pl}}| {f}_{\mathrm{cell}},{n}_{\mathrm{trial}})\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&=\,{{Cf}}_{\mathrm{cell}}^{{n}_{\mathrm{pl}}}{(1-{f}_{\mathrm{cell}})}^{{n}_{\mathrm{nd}}},\end{eqnarray} \tag{ 9 }$$

where C is the following normalization constant:¹⁶

$$\begin{eqnarray}&&C=\displaystyle \frac{({n}_{\mathrm{trial}}+1){\rm{\Gamma }}({n}_{\mathrm{trial}}+1)}{{\rm{\Gamma }}({n}_{\mathrm{pl}}+1){\rm{\Gamma }}({n}_{\mathrm{nd}}+1)}.\end{eqnarray} \tag{ 10 }$$

If no transits are detected in a given cell, we may place an upper bound on the occurrence rate by calculating the maximum value of ${f}_{\mathrm{cell}}$ that yields zero planet detections (${n}_{\mathrm{pl}}=0$) with 90% probability. This is done by numerically finding an ${f}_{\mathrm{cell}}$ that satisfies

$$\begin{eqnarray}&&{\int }_{0}^{{f}_{\mathrm{cell}}}P(f| {n}_{\mathrm{pl}},0){df}=90 \% .\end{eqnarray} \tag{ 11 }$$

When analyzing planet occurrence as a function of metallicity, we must consider the fraction of stars in ${ \mathcal S }$ that falls within a specified range of metallicity. Therefore, to compute ${f}_{\mathrm{cell}}$ for a cell bounded by $[{R}_{P,1},{R}_{P,2}]$, $[{P}_{1},{P}_{2}]$, and $[{M}_{1},{M}_{2}]$, we simply multiply ${n}_{\mathrm{trial}}$ from Equation (7) by the fraction of LAMOST stars with metallicities between ${M}_{1}$ and ${M}_{2}$. This treatment assumes that planet detectability does not depend on metallicity. We justify this assumption in Appendix B.

To compute occurrence, we must estimate $\langle {p}_{\mathrm{tr}}{p}_{\det }\rangle$. The probability that a randomly inclined planet at a distance of a will transit with b < 0.9  is ${p}_{\mathrm{tr}}$ = 0.9  ${R}_{\star }$/a, assuming circular orbits. We compute p_tr for each star using the tabulated values of ${R}_{\star }$ and ${M}_{\star }$ from the Q1–Q17 (DR25) stellar properties table (Mathur et al. 2017), along with Kepler's Third Law.

The probability of detecting a transiting planet with size ${R}_{P}$ and period P depends on the number of factors, the most significant of which is the S/N. We follow the methodology of Fulton et al. (2017) and compute an expected S/N using the following formula:

$$\begin{eqnarray}&&{\rm{S}}/{\rm{N}}={\left(\displaystyle \frac{{R}_{P}}{{R}_{\star }}\right)}^{2}\sqrt{\displaystyle \frac{{T}_{\mathrm{obs}}}{P}}\left(\displaystyle \frac{1}{\sigma ({T}_{14})}\right).\end{eqnarray} \tag{ 12 }$$

Here, T_obs is the time the star was observed by Kepler, and $\sigma ({T}_{14})$ represents the photometric variability on transit timescales. The DR25 stellar properties table lists T_obs, ${R}_{\star }$, and photometric noise¹⁷ on 3, 6, and 12 hr timescales. We compute $\sigma ({T}_{14})$ at intermediate values of T₁₄ through piecewise linear interpolation (or extrapolation) in log σ–logT₁₄ space.

Characterizing ${p}_{\det }$ as a function of S/N has been addressed in several previous works (e.g., Fressin et al. 2013; Petigura et al. 2013; Christiansen et al. 2015). This is a challenging problem, which depends on how a complex, multi-stage transit pipeline performs in the face of correlated, non-stationary, and non-Gaussian photometric noise. We adopted the ${p}_{\det }$ (S/N) of Fulton et al. (2017), who used the transit injections of Christiansen et al. (2015) to derive the following relationship:

$$\begin{eqnarray}&&{p}_{\det }(s)={\rm{\Gamma }}(k){\int }_{0}^{\tfrac{s-l}{\theta }}{t}^{k-1}{e}^{-t}{dt},\end{eqnarray} \tag{ 13 }$$

where s is the S/N, k = 17.56, l = 1.00, and $\theta =0.49$. We illustrate $\langle {p}_{\det }\rangle$ and $\langle {p}_{\mathrm{tr}}{p}_{\det }\rangle$ as a function of P and ${R}_{P}$ in Figure 5.

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We often wish to characterize the occurrence distribution with parametric models. These models serve to quantify shapes and trends in the population of planets. We extend the maximum-likelihood method of Howard et al. (2012) to fit ${Cg}({\boldsymbol{x}},{\boldsymbol{z}};{\boldsymbol{\theta }})$ to the observed planet population, where ${\boldsymbol{\theta }}$ represents the vector of shape parameters in our model.

The occurrence within each cell ${f}_{\mathrm{cell}}$ gives an estimate of the differential occurrence rate via

$$\begin{eqnarray}&&{Cg}({\boldsymbol{x}},{\boldsymbol{z}};{\boldsymbol{\theta }})=\displaystyle \frac{{f}_{\mathrm{cell}}}{{\rm{\Delta }}{\boldsymbol{x}}}.\end{eqnarray} \tag{ 14 }$$

The log-likelihood of a given model with parameters $\{C,{\boldsymbol{\theta }}\}$ given the observed rate in a single cell labeled by i is

$$\begin{eqnarray}&&\mathrm{ln}{L}_{i}={n}_{\mathrm{pl},i}\mathrm{ln}{Cg}{\rm{\Delta }}{\boldsymbol{x}}+{n}_{\mathrm{nd},i}\mathrm{ln}(1-{Cg}{\rm{\Delta }}{\boldsymbol{x}}).\end{eqnarray} \tag{ 15 }$$

Each cell is an independent constraint on ${Cg}({\boldsymbol{x}},{\boldsymbol{z}};{\boldsymbol{\theta }})$, so a fit to multiple cells requires maximizing the combined log-likelihood over all cells:

$$\begin{eqnarray}&&\mathrm{ln}L=\displaystyle \sum _{i=1}^{{n}_{\mathrm{cell}}}{L}_{i}.\end{eqnarray} \tag{ 16 }$$

When fitting parameterized models of occurrence in Sections 5 and 6, we use very small bins having ${\rm{\Delta }}\mathrm{log}P\,=0.05\,\mathrm{dex}$ and ${\rm{\Delta }}\mathrm{log}M=0.05\,\mathrm{dex}$. As a result, many bins have one or zero planet detections. The maximum-likelihood method is stable for small bin sizes because Equation (15) incorporates non-detections. We experimented with even finer bins, but did not observe significant changes in the results. After finding the max-likelihood model, we explore the credible range of models using Markov Chain Monte Carlo (MCMC) sampling.¹⁸

We found that metallicity is associated with the prevalence of large planets. This trend is consistent with the planet-metallicity correlation for Jovian-mass planets found by RV surveys (Santos et al. 2004; Fischer & Valenti 2005) and with previous analyses of the metallicities of Kepler planet hosts (e.g., Buchhave et al. 2012). We observed that for smaller planets, the metallicity correlation weakens, which is consistent with trends observed in previous RV studies (e.g., Sousa et al. 2008; Ghezzi et al. 2010) and metallicity studies of Kepler planet hosts (e.g., Buchhave et al. 2012, 2014). Our analysis builds upon these previous studies by providing planet occurrence as a function of metallicity based on the high-precision, high purity CKS catalog of planet radii and metallicities.

There are roughly 15 warm super-Earths per 100 stars over the metallicities ranging from −0.4 to +0.4 dex. If we assume that stellar metallicity traces disk solid surface densities from 0.10 to 0.40 au (which correspond P ≈ 10–100 day orbits), then warm super-Earths can form with moderate efficiency, even in disks with low solid surface densities. Moreover, a factor of 6 increase in solids does not enhance the likelihood of a star to host a warm super-Earth.

RV and TTV mass measurements of super-Earths typically find masses of 1–10 ${M}_{\oplus }$ (Marcy et al. 2014, see Figure 7 of Sinukoff et al. 2017 for an updated census). The bulk densities of these planets are usually consistent with combinations of rock and iron, but are inconsistent with even small envelope fractions of ${M}_{\mathrm{env}}/{M}_{P}\sim 1 \%$, which would significantly reduce the observed bulk densities.

Even though sub-Neptunes are significantly larger than super-Earths, they share several important characteristics. Sub-Neptunes are typically 5–15 ${M}_{\oplus }$, which overlaps with the super-Earth mass range. Sub-Neptune bulk densities require H/He envelopes that are 1%–10% of the planet's mass, which significantly enlarge the planets without significantly altering their masses. Therefore, the typical sub-Neptune contains a comparable amount of solid material as a typical super-Earth. Also, their overall occurrence rates are roughly comparable at 5%–10% per 0.25 dex period interval (Figure 9).

We see an important difference between super-Earths and sub-Neptunes when we compare their dependencies on stellar metallicity. There are comparable numbers of warm super-Earths and warm sub-Neptunes around stars having $[\mathrm{Fe}/{\rm{H}}]$ = [−0.4, −0.2] dex, about 20 per 100 stars. Unlike the warm super-Earths, the occurrence of warm sub-Neptunes increases with metallicity. There are about 40 warm sub-Neptunes per 100 stars having $[\mathrm{Fe}/{\rm{H}}]$ = [+0.2, +0.4] dex.

Perhaps the enriched disks of metal-rich stars form super-Earths more efficiently, but they easily acquire the few percent envelopes and are converted into sub-Neptunes. This could occur if super-Earths form more quickly in metal-rich disks, allowing for more time to accrete 1%–10% gaseous envelopes.

Such a timing argument was proposed by Dawson et al. (2015), to explain an apparent absence of warm super-Earths around the highest metallicity stars ($[\mathrm{Fe}/{\rm{H}}]$ > 0.18 dex) in the Buchhave et al. (2014) metallicity catalog.²⁰ While only 12/132 (9%) of our warm super-Earths have $[\mathrm{Fe}/{\rm{H}}]$ > 0.18 dex, this is consistent with the fact that only 12% of field stars have $[\mathrm{Fe}/{\rm{H}}]$ > 0.18 dex. As shown in Figures 4 and 10, we do not observe such an absence of warm super-Earths, but instead find nearly equal occurrence of warm super-Earths around stars of wide-ranging metallicities. This underscores the importance of having accurate knowledge of the distribution of field star metallicities in planet-metallicity studies. Nevertheless, accelerated formation of cores in metal-rich disks may explain the rise in warm sub-Neptune occurrence with metallicity.

An alternative to the timing argument, discussed above, is that high disk metallicities may somehow increase the rate of envelope accretion. However, the gas accretion models of Pollack et al. (1996) and (more recently) Lee & Chiang (2015) predict the opposite effect: dusty envelopes should accrete more slowly, due to higher opacities and longer cooling timescales.

As we consider larger warm planets, the metallicity correlation becomes stronger. Sub-Saturns have a β index of $+{2.1}_{-0.7}^{+0.7}$. If disk metallicity assists with the accretion of gas through accelerated core formation, it may also explain the sub-Saturns. However, abrupt strengthening of the metallicity correlation above 4 ${R}_{\oplus }$ and the roughly order of magnitude decrease in the frequency of sub-Saturns compared to sub-Neptunes suggests the formation pathways of sub-Saturns and sub-Neptunes may be quite different.

Around 20 sub-Saturns have well-measured masses from either RVs or TTVs (see Petigura et al. 2017b for a recent compilation). For sub-Saturns, we observe an order of magnitude scatter in the observed masses at a given size, indicating a diversity in core and envelope masses. While some sub-Saturns have ≈5 ${M}_{\oplus }$ cores, similar to the super-Earths and sub-Neptunes, many have cores of ≈50 ${M}_{\oplus }$. In addition, the most massive sub-Saturns tend to be found around the most metal-rich hosts (Petigura et al. 2017b).

The existence of ≈50 ${M}_{\oplus }$ cores in planets with 20% envelope fractions challenges the classic core-accretion models of Pollack et al. (1996) that predict that cores larger than 10 ${M}_{\oplus }$ should undergo runaway accretion. Perhaps these massive sub-Saturn cores are the result of the late-stage mergers (or series of mergers) of 10 ${M}_{\oplus }$ cores. This formation scenario requires one or more closely spaced 10 ${M}_{\oplus }$ planets. As we have shown, the probability for a star to produce a 10 ${M}_{\oplus }$ core increases with metallicity. Therefore, the probability for a star to produce two 10 ${M}_{\oplus }$ cores likely increases with a steeper power law index. Thus, the production of sub-Saturns by collisions may explain the mass–metallicity dependence and the steeper relationship between metallicity and planet occurrence. This could also explain the mass–metallicity correlation. This theory also predicts that if late-stage mergers play a large role in the formation of sub-Saturns, they should produce relic eccentricities that are observable at later times.

Our planet sample includes only four warm Jupiters, which is insufficient to search for trends. Fischer & Valenti (2005) analyzed 1040 FGK-type stars from the Keck, Lick, and Anglo-Australian Telescope planet search programs and found a planet-metallicity correlation with a $\beta =2$ index. However, this sample included planets with P = 1–4000 days, with the bulk of the sample having $P\gt 300\,\mathrm{days}$ longer than the periods considered here.

We found that metallicity is associated with the presence of short-period planets having P < 10 days. This is consistent with trends observed by Mulders et al. (2016) and Dong et al. (2018), who studied the host star metallicities for samples of 665 and 295 planets, respectively. Both studies used spectroscopically constrained metallicities from LAMOST. Our analysis incorporates the high-precision CKS metallicities for a larger sample of  970  planets.

While close-in planets, of all sizes, are rare compared to more distant planets, their frequency increases as a function of stellar metallicity. In Section 5, when considering just the period distribution of super-Earths and sub-Neptunes, we found that for $P\lt 10\,\mathrm{days}$, the occurrence of super-Earths and sub-Neptunes declines with decreasing period. The slope of this falloff is approximated by ${df}\propto {P}^{\alpha }d\mathrm{log}P$, with α values of ${2.4}_{-0.3}^{+0.4}$ and ${2.3}_{-0.2}^{+0.2}$, respectively.

In Section 6, when considering the joint P–$[\mathrm{Fe}/{\rm{H}}]$ occurrence distribution, we found that both hot super-Earths and sub-Neptunes are correlated with host star metallicity. This is especially noteworthy for the hot super-Earths, given that the warm super-Earths exhibit no such correlation. For $P\lt 10\,\mathrm{days}$, the frequency of hot super-Earths grows from 4% for stars having $[\mathrm{Fe}/{\rm{H}}]$ = [−0.4, −0.2] dex to 10% for stars having $[\mathrm{Fe}/{\rm{H}}]$ = [+0.2, +0.4] dex (see Figure 10). We see an even steeper increase in the hot sub-Neptune rates from 1% to 8% for the same metallicity intervals. In Sections 7.2.1 and 7.2.2, we consider two possible formation pathways: (1) in situ models, where planets form near their final location, and (2) high eccentricity migration, where the semimajor axes of fully formed planets are altered by major scattering events. In Section 7.2.3, we consider some observational tests that may distinguish between these scenarios.

7.2.1. In Situ Formation

7.2.2. High Eccentricity Migration

7.2.3. Observational Tests

Hot Jupiters represent an extreme of planet formation, and many theories have been proposed to explain their formation. A non-exhaustive list of theories includes star–planet Kozai (e.g., Wu & Murray 2003), planet–planet Kozai (e.g., Naoz et al.2011), smooth disk-driven Type-I migration (e.g., Ida & Lin 2008), high eccentricity migration, and in situ formation (e.g., Batygin et al. 2016). Hot Jupiters are also of interest because they offer a point of comparison between the Kepler population of planets and those around nearby stars.

We find a hot Jupiter rate of  ${0.57}_{-0.12}^{+0.14}$ %,²¹ which is about $1.5\sigma$ larger than 0.4 ± 0.1% reported by Howard et al. (2012). This difference cannot be due to differences the treatment of completeness; for hot Jupiters, ${p}_{\det }$ is nearly unity.

The larger occurrence rate presented here is likely due to the improved radius measurements in the CKS sample. Howard et al. (2012) included 13 hot planets with ${R}_{P}$ = 5.6–8.0 ${R}_{\oplus }$, slightly too small to be classified as a hot Jupiter by their criteria. With our improved CKS planet radii, we have found that these large, hot sub-Saturns are exceptionally rare. Our planet sample ${ \mathcal P }$ includes just 3 such planets, while Howard et al. (2012) included 13, even though their stellar parent is less than twice as large as ours ${n}_{{ \mathcal S }}$ (58,041 versus  33,020). Some of the hot sub-Saturns in Howard et al. (2012) were misidentified hot Jupiters.

A major challenge in comparing the hot Jupiter rates in the Kepler and the solar neighborhood is their low intrinsic occurrence. Even though our planet sample ${ \mathcal P }$ was drawn from a parent stellar sample ${ \mathcal S }$ containing  33,020  stars, ${ \mathcal P }$ contained just  14  detections due to low intrinsic occurrence and the requirement of transiting geometries. Even blind RV surveys which are not limited to transiting geometries must observe hundreds of stars to detect a few hot Jupiters. Another challenge is that RV surveys are typically not magnitude-limited samples like ${ \mathcal S }$. RV surveys often prioritize stars that are single, slowly rotating, and have stellar properties that fall within a preselected range of interest (e.g., M-stars or evolved stars). Wright et al. (2012) attempted to retroactively remove these selection effect from the California Planet Search database, to produce a magnitude-limited sample of stars that could be directly compared to the Kepler stellar sample. Wright et al. (2012) reported 10 hot Jupiters from a sample of 836 stars or a rate of 1.2 ± 0.4%, twice the rate presented here. However, the large fractional uncertainty in the Wright et al. (2012) measurement means that these two results differ by only 1.5σ.

In Section 6, we modeled the hot Jupiter occurrence as ${df}\propto {P}^{\alpha }{10}^{\beta M}d\mathrm{log}{PdM}$ and found a β index of $+{3.4}_{-0.8}^{+0.9}$. This strong association between hot Jupiters and metallicity has also been observed by RV stars. Fischer & Valenti (2005) studied the occurrence of Jupiter-mass planets in the CPS sample and modeled their occurrence according to ${df}\propto {10}^{\beta M}$ and β index of 2.0. Johnson et al. (2010) performed a similar analysis, considered an additional mass dependence ${df}\propto {M}_{\star }^{\alpha }{10}^{\beta M}$ and found a β index of 1.2 ± 0.2. It is worth recalling that both studies included long period planets ($P\gt 1\,\mathrm{year}$), which constituted the bulk of the sample. Guo et al. (2017) repeated the Johnson et al. (2010) analysis, but restricted the study to hot Jupiters and found β = 2.1 ± 0.7. While this result differs with our measurement of β by about 2σ, it is clear that hot Jupiters in the solar neighborhood and in the Kepler field are both strongly associated with metallicity.

In the previous sections, we have noted the general tendency for the metallically correlation to steepen with decreasing orbital period and increasing planet size, and have hypothesized some processes that could account for these trends such as accelerated core/envelope growth, higher disk densities, and planet–planet scattering. Given this general trend, it is perhaps not surprising that hot Jupiters would have the strongest metallicity correlation. However, given that the hot Jupiters occupy a distinct island on the P–${R}_{P}$ plane, it is possible that their formation pathway is radically different than that of the other planet classes considered here.

Our target sample contains very few stars (16) with metallicities below −0.4 dex. As a result, we are insensitive to planet occurrence at low metallicities. Probing planet occurrence at low metallicities would shed light on some of the questions raised by this analysis, such as, What is the minimum metallicity needed to form a warm super-Earth?

Since metal-poor stars are rare, a blind survey like Kepler must target a large number of stars in order to capture sufficient numbers of low metallicity stars. There is some room for improvement with the existing Kepler sample. Our sample was drawn from ≈38,000 of the ≈150,000 stars observed by Kepler due to our magnitude limit. Characterizing the metallicities of Kepler planet hosts and field stars fainter than ${Kp}$ = 14.2 mag would enlarge the sample of metal-poor stars. However, given that these new stars would be between ${Kp}$ = 14–16 mag, their amenability to the detection of small planets is poor.

Upcoming missions like TESS (Ricker et al. 2015) and PLATO (Rauer et al. 2014) will survey more stars than Kepler and thus cast a significantly larger net for rare metal-poor stars. Gaia will also identify metal-poor stars across the sky. A targeted RV survey of low metallicity stars would also probe planet occurrence at low metallicities. Such a survey would not be restricted to transiting planets, and would need to survey fewer stars to gather statistically significant planet samples.

Finally, one could search for planets in globular clusters. Gilliland et al. (2000) conducted an important early transit search with HST, which targeted 47 Tucanae ($[\mathrm{Fe}/{\rm{H}}]$ = −0.78 dex). The expected yield was 17 hot Jupiters, but the survey yielded no detections, which was attributed to low metallicities. Recently, Masuda & Winn (2017) re-assessed the significance of the null result, using updated knowledge of the typical size, periods, and host star properties of hot Jupiters. Their revised yield calculation was 2 ± 1, and thus the null result does not require a metallicity effect. However, based on the steep dependence of hot Jupiter occurrence with metallicity, yields would likely be low. Future globular cluster surveys with higher precision or longer baselines would help constrain planet population at −1 dex. However, high stellar densities of globulars may complicate direct comparisons to field stars.